its side, and goods are to be removed from the vessels into the upper stories. Instead of removing the goods into the store, and hoisting them in the direction of a, it is only necessary to carry the rope b, over the pulley c, which is at the end of a strong beam projecting out from the side of the store, and then the goods will be raised in the direction of d, thus saving the labor of moving them twice.

The wheel and axle, under different forms, is applied to a variety of common purposes. Fig. 57.

The capstan, in universal use, or board of ships and other vessels, is an axle placed upright, with a head, bor drum, a, fig. 57, pierced with

holes, for the levers b, c, d. The c weight is drawn by the rope e, passing two or three times round the axle to prevent its slipping.

This is a very powerful, and convenient machine. When not in use, the levers are taken out of their pla ces and laid aside, and when great force is required, two, or three men can push at each lever.

The common windlass for drawing water, is another modi. fication of the wheel and axle. The winch, or crank, by which it is turned, is moved around by the hand, and there is no difference in the principle, whether a whole wheel is turned, or a single spoke. The windlass, therefore, answers to the wheel, while the rope is taken up, and the weight raised by the axle, as already described. Fig. 58.

In cases where great weights are to be raised, and it is required that the machine should be as small as possible, on ac. count of room, the simple wheel and axle, modified as represented by fig. 58, is sometimes used.

The axle may be considered in two parts, one

What is the capstan, where is it chiefly used? What are the peculiar advantages of this form of the wheel and axle ? In the common windlass, what part answers to the wheel?

of which is larger than the other. The rope is attached by its two ends to the ends of the axle, as seen in the figure. The weight to be raised is attached to a small pulley, or wheel, round which the rope passes. The elevation of the weight may be thus described. Upon turning the axle, the rope is coiled around the larger part, and at the same time it is thrown off the smaller part. At every revolution, therefore, a portion of the rope will be drawn up, equal to the circumference of the thicker part, and at the same time a portion, equal to that of the thinner part, will be let down. On the whole, then, one revolution of the machine will shorten the rope where the weight is suspended, just as much as the difference between the circumference of the two parts.

Fig. 59.

h

Now to understand the principle on which this machine acts, we must refer to fig. 59, where it is obvious that the two parts of the rope a and b, equally support the weight d, and that the rope, as the machine turns, passes from the small part of the axle e, to the large part h, consequently the weight does not rise in a perpendicular line towards c, the centre of both, but in a line between the outsides of the large and small parts. Let us consider what would be the consequence of changing the rope a to the larger part of the a axle, so as to place the weight in a line per

pendicular to the axis of motion. In this case, it is obvious that the machine would be in equilibrium, since the weight d, would be divided between the two sides equally, and the two arms of a lever passing through the centre c, would be of equal length, and therefore no advantage would be gained. But in the actual arrangement, the weight being sustained equally by the large and small parts, there is involved a lever power, the long arm of which is equal to half the diameter of the large part, while the short arm is equal to half the diameter of the small part, the fulcrum being between them.

Explain fig. 58. Why is the rope shortened, and the weight raised? What is the design of fig. 59? Does the weight rise perpendicular to the axis of motion? Suppose the cylinder was, throughout, of the same size, what would be the consequence? On what principle does this machine act? Which are the long, and short arms of the lever, and where is the fulcrum?

As the wheel and axle is only a modification of the simple lever, so a system of wheels acting on each other, and transmitting the power to the resistance, is only another form of the compound lever.

arm from the same centre to the ends of the cogs. The dotted line c, passing through the centre of the wheel a, shows the position of the lever, as the wheel now stands. The centre on which both wheels turn, it will be obvious, is the fulcrum of this lever. As the wheel turns, the short arm of this lever will act upon the long arm of the next lever by means of the teeth on the circumference of the wheel b, and this again through the teeth on the axle of b, will transmit its force to the circumference of the wheel d, and so by the short arm of the third lever to the weight w. As the power, or small weight falls, therefore, the resistance, w, is raised, with the multiplied force of three levers, acting on each other.

In respect to the force to be gained by such a machine, suppose the number of teeth on the axle of the wheel a, to be six times less than the number of those on the circumference of the wheel b, then b would only turn round once, while a turned six times. And in like manner, if the number of teeth on the circumference of d, be six times greater than those on the axle of b, then d would turn once, while b turned six times. Thus six revolutions of a would make b revolve once, and six revolutions of b, would make d revolve once. Therefore a makes thirty-six revolutions, while d makes only one.

On what principle does a system of wheels act, as represented in fig. 60? Explain fig. 60, and show how the power, p, is transferred, by the action of levers to w.

The diameter of the wheel a, being three times the diame. ter of the axle of the wheel d, and its velocity of motion being 36 to 1, 3 times 36 will give the weight which a power of 1 pound at p, would raise at w. Thus 36×3=108. pound at p would therefore balance 108 pounds at w.

One

If the student has attended closely to what has been said on mechanics, he will now be prepared to understand, that no machine, however simple or complex it may be, can create the least degree of force. It is true that one man with a machine, may apply a force which a hundred could not exert with their hands, but then it would take him a hundred times as long.

Suppose there are twenty blocks of stone to be moved a hundred feet; perhaps twenty men, by taking each a block, would move them all in a minute. One man, with a capstan, we will suppose, may move them all at once, but this man, with his lever, would have to make one revolution for every foot he drew the whole load towards him, and therefore to make one hundred revolutions to perform the whole work. It would also take him twenty times as long to do it, as it took the twenty men. His task, indeed, would be more than twenty times harder than that performed by the twenty men, for in addition to moving the stone, he would have the friction of the machinery to overcome, which commonly amounts to nearly one third of the force employed.

Hence there would be an actual loss of power by the use of the captan, though it might be a convenience for the one man to do his work by its means, rather than to call in nineteen of his neighbours to assist him.

The same principle holds good in respect to other machi. nery, where the strength of man is employed as the power, or prime mover. There is no advantage gained, except that of convenience. In the use of the most simple of all machines, the lever, and where, at the same time, there is the least force lost by friction, there is no actual gain of power, for what seems to be gained in force is always lost in velocity. Thus if a lever is of such length to raise 100 pounds an inch by the power of one pound, its long arm must pass through a

What weight will one pound at p, balance at w? Is there any actual power gained by the use of machinery? Suppose 20 men to move 20 stones to a certain distance with their hands, and one man moves them back to the same place with a capstan, which performs the most actual labor? Why? Why then is machinery a convenience?

space of 100 inches. Thus what is gained in one way is lost in another.

Any power by which a machine is moved, must be equal to the resistance to be overcome, and, in all cases where the power descends, there will be a proportion between the velocity with which it moves downwards, and the velocity with which the weight moves upwards. There will be no differ

ence in this respect, whether the machine be simple or compound, for if its force be increased by increasing the number of levers, or wheels, the velocity of the moving power must also be increased, as that of the resistance is diminished.

There being, then, always a proportion, between the velocity with which the moving force descends, and that with which the weight ascends, whatever this proportion may be, it is necessary that the power should have to the resistance the same ratio that the velocity of the resistance has to the velocity of the power. In other words, "The power multiplied by the space through which it moves, in a vertical direction, must be equal to the weight multiplied by the space through which it moves in a vertical direction."

This law is known under the name of "the law of virtual velocities," and is considered the golden rule of mechanics. This principle has already been explained, while treating of the lever, but that the student should want nothing to assist him in clearly comprehending so important a law, we will again illustrate it in a different manner.

b

Fig. 61.

a

Suppose a weight of ten pounds to be suspended on the short arm of the lever, fig. 61, and that the fulcrum is only one inch from the weight; then, if the lever be ten inches long, on the other side of the fulcrum, one pound at a would raise, or balance, the ten pounds at b. But in raising the ten pounds one inch in a vertical direction, the long arm of the lever would fall

In the use of the lever, what proportion is there between the force of the short arm, and the velocity of the long arm? How is this illustrated? It is said, that the velocity of the power downwards, must be in proportion to that of the weight upwards? Does it make any difference, in this respect, whether the machine be simple or compound? What is the golden rule of mechanics? Under what name is this law known? Explain fig. 61, and show how the rule is illustrated by that figure.